Let X ~ Geom(p), let n be a non-negative integer, and let Y ~ Bin(n, p). Show that P(X = n) = (l/n)(1/n)p(Y=1).

Solution :

Step 1 of 1:

Given and

Where mean and variance .

Here non-negative integer is n.

Our goal is :

We need to prove that P(X=n)=.

Now we have to prove that P(X=n)=.

P(X=n)=

The formula of the geometric distribution is

P(X=n)=

Where q = (1-p)

So P(X=n)=

Then,

P(X=n)=

The formula of the binomial distribution is

Here the probability of X is n and the probability of Y is 1.

P(X=n)=

Here and = p.

P(X=n)=

Therefore P(X=n)=.